Page 1 EXERCISES For JEE Main Subjective Questions Newton's Law of Gravitation Q 1. Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles. Q 2. Three particles A, B and C, each of mass m, are placed in a line with AB = BC = d. Find the gravitational force on a fourth particle P of same mass, placed at a distance d from the particle B on the perpendicular bisector of the line AC. Q 3. Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre. Q 4. Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Q 5. The figure shows a uniform rod of length l whose mass per unit length is ? . What is the gravitational force of the rod on a particle of mass m located a distance d from one end of the rod? Acceleration due to Gravity Q 6. Value of g on the surface of earth is 9.8 m/s 2 . Find its value on the surface of a planet whose mass and radius both are two times that of earth. Q 7. Value of g on the surface of earth is 9.8 m/s 2 . Find its value : (a) at height h = R from the surface, (b) at depth h = R 2 from the surface. (R = radius of earth) Q 8. Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below and above the surface of the earth. Q 9. A body is weighed by a spring balance to be 1000 N at the north pole. How much will it weight at the equator. Account for the earth's rotation only. Q 10. At what rate should the earth rotate so that the apparent g at the equator becomes zero ? What will be the length of the day in this situation ? Q 11. Assuming earth to be spherical, at what height above the north pole, value of g is same as that on the earth's surface at equator ? Gravitational Field Strength and Potential Page 2 EXERCISES For JEE Main Subjective Questions Newton's Law of Gravitation Q 1. Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles. Q 2. Three particles A, B and C, each of mass m, are placed in a line with AB = BC = d. Find the gravitational force on a fourth particle P of same mass, placed at a distance d from the particle B on the perpendicular bisector of the line AC. Q 3. Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre. Q 4. Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Q 5. The figure shows a uniform rod of length l whose mass per unit length is ? . What is the gravitational force of the rod on a particle of mass m located a distance d from one end of the rod? Acceleration due to Gravity Q 6. Value of g on the surface of earth is 9.8 m/s 2 . Find its value on the surface of a planet whose mass and radius both are two times that of earth. Q 7. Value of g on the surface of earth is 9.8 m/s 2 . Find its value : (a) at height h = R from the surface, (b) at depth h = R 2 from the surface. (R = radius of earth) Q 8. Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below and above the surface of the earth. Q 9. A body is weighed by a spring balance to be 1000 N at the north pole. How much will it weight at the equator. Account for the earth's rotation only. Q 10. At what rate should the earth rotate so that the apparent g at the equator becomes zero ? What will be the length of the day in this situation ? Q 11. Assuming earth to be spherical, at what height above the north pole, value of g is same as that on the earth's surface at equator ? Gravitational Field Strength and Potential Q 12. Two concentric spherical shells have masses m 1 ,m2 and radii R1, R2 (R1 < R2). Calculate the force exerted by this system on a particle of mass m, if it is placed at a distance 12 (R R ) 2 ? from the centre. Q 13. Two spheres one of mass M has radius R. Another sphere has mass 4M and radius 2R. The centre to centre distance between them is 12R. Find the distance from the centre of smaller sphere where : (a) net gravitational field is zero, (b) net gravitational potential is half the potential on the surface of larger sphere. Q 14. A semicircular wire has a length L and mass M. Find the gravitational field at the centre of the circle. Q 15. A uniform solid sphere of mass M and radius a is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2a. Find the gravitational field at a distance (a) 3 a 2 from the centre, (b) 5 a 2 from the centre. Q 16. The density inside a solid sphere of radius a is given by ? = ?0a/r, where ?0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre. Q 17. A particle of mass m is placed at the centre of a uniform spherical shell of same mass and radius R. Find the gravitational potential at a distance R 2 from the centre. Gravitational Potential Energy Q 18. A rocket is accelerated to speed v 2 gR ? near earth's surface (R =radius of earth). Show that very far from earth its speed will be v 2gR ? . Q 19. Two neutron stars are separated by a distance of 10 10 m. They each have a mass of 10 30 kg and a radius of 10 5 m. They are initially at rest with respect to each other. As measured from the rest frame, how fast are they moving when : (a) their separation has decreased to onehalf its initial value, (b) they are about to collide. Q 20. A projectile is fired vertically from earth's surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of earth will it go ? Q 21. A mass m is taken to a height R from the surface of the earth and then is given a vertical velocity v. Find the minimum value of v, so that mass never returns to the surface of the earth. (Radius of earth is R and mass of the earth M). Q 22. In the figure masses 400 kg and 100 kg are fixed. Page 3 EXERCISES For JEE Main Subjective Questions Newton's Law of Gravitation Q 1. Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles. Q 2. Three particles A, B and C, each of mass m, are placed in a line with AB = BC = d. Find the gravitational force on a fourth particle P of same mass, placed at a distance d from the particle B on the perpendicular bisector of the line AC. Q 3. Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre. Q 4. Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Q 5. The figure shows a uniform rod of length l whose mass per unit length is ? . What is the gravitational force of the rod on a particle of mass m located a distance d from one end of the rod? Acceleration due to Gravity Q 6. Value of g on the surface of earth is 9.8 m/s 2 . Find its value on the surface of a planet whose mass and radius both are two times that of earth. Q 7. Value of g on the surface of earth is 9.8 m/s 2 . Find its value : (a) at height h = R from the surface, (b) at depth h = R 2 from the surface. (R = radius of earth) Q 8. Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below and above the surface of the earth. Q 9. A body is weighed by a spring balance to be 1000 N at the north pole. How much will it weight at the equator. Account for the earth's rotation only. Q 10. At what rate should the earth rotate so that the apparent g at the equator becomes zero ? What will be the length of the day in this situation ? Q 11. Assuming earth to be spherical, at what height above the north pole, value of g is same as that on the earth's surface at equator ? Gravitational Field Strength and Potential Q 12. Two concentric spherical shells have masses m 1 ,m2 and radii R1, R2 (R1 < R2). Calculate the force exerted by this system on a particle of mass m, if it is placed at a distance 12 (R R ) 2 ? from the centre. Q 13. Two spheres one of mass M has radius R. Another sphere has mass 4M and radius 2R. The centre to centre distance between them is 12R. Find the distance from the centre of smaller sphere where : (a) net gravitational field is zero, (b) net gravitational potential is half the potential on the surface of larger sphere. Q 14. A semicircular wire has a length L and mass M. Find the gravitational field at the centre of the circle. Q 15. A uniform solid sphere of mass M and radius a is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2a. Find the gravitational field at a distance (a) 3 a 2 from the centre, (b) 5 a 2 from the centre. Q 16. The density inside a solid sphere of radius a is given by ? = ?0a/r, where ?0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre. Q 17. A particle of mass m is placed at the centre of a uniform spherical shell of same mass and radius R. Find the gravitational potential at a distance R 2 from the centre. Gravitational Potential Energy Q 18. A rocket is accelerated to speed v 2 gR ? near earth's surface (R =radius of earth). Show that very far from earth its speed will be v 2gR ? . Q 19. Two neutron stars are separated by a distance of 10 10 m. They each have a mass of 10 30 kg and a radius of 10 5 m. They are initially at rest with respect to each other. As measured from the rest frame, how fast are they moving when : (a) their separation has decreased to onehalf its initial value, (b) they are about to collide. Q 20. A projectile is fired vertically from earth's surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of earth will it go ? Q 21. A mass m is taken to a height R from the surface of the earth and then is given a vertical velocity v. Find the minimum value of v, so that mass never returns to the surface of the earth. (Radius of earth is R and mass of the earth M). Q 22. In the figure masses 400 kg and 100 kg are fixed. (a) How much work must be done to move a 1 kg mass from point A to point B ? (b) What is the minimum kinetic energy with which the 1 kg mass must be projected from A to the right to reach the point B ? Planets and Satellites : Kepler's Law Q 23. A sky lab of mass 2 x 10 3 kg is first launched from the surface of earth in a circular orbit of radius 2R and then it is shifted from this circular orbit to another circular orbit of radius 3R. Calculate the energy required: (a) to place the lab in the first orbit, (b) to shift the lab from first orbit to the second orbit. (R = 6400 km, g = 10m/s 2 ) Q 24. Two identical stars of mass M orbit around their centre of mass. Each orbit is circular and has radius R, so that the two stars are always on opposite sides of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) What minimum energy would be required to separate the two stars to infinity ? Q 25. Consider two satellites A and B of equal mass, moving in the same circular orbit of radius r around the earth but in the opposite sense and therefore a collision occurs. (a) Find the total mechanical energy EA+EB of the two satelliteplusearth system before collision. (b) If the collision is completely inelastic, find the total mechanical energy immediately after collision. Describe the subsequent motion of the combined satellite. Q 26. Two satellites A and B revolve around a planet in two coplanar circular orbits in the same sense with radii 10 4 km and 2 x 10 4 km respectively. Time period of A is 28 hours. What is time period of another satellite. Q 27. A satellite of mass 1000 kg is supposed to orbit the earth at a height of 2000 km above the earth's surface. Find (a) its speed in the orbit, (b) its kinetic energy, (c) the potential energy of the earthsatellite system and (d) its time period. Mass of the earth = 6 x 10 24 kg. Q 28. In a certain binary star system, each star has the same mass as our sun. They revolve about their centre of mass. The distance between them is the same as the distance between earth and the sun. What is their period of revolution in years ? Q 29. (a) Does it take more energy to get a satellite upto 1500 km above earth than to put it in circular orbit once it is there. (b) What about 3185 km? (c) What about 4500 km? (Take Re = 6370 km) Solutions 1. Page 4 EXERCISES For JEE Main Subjective Questions Newton's Law of Gravitation Q 1. Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles. Q 2. Three particles A, B and C, each of mass m, are placed in a line with AB = BC = d. Find the gravitational force on a fourth particle P of same mass, placed at a distance d from the particle B on the perpendicular bisector of the line AC. Q 3. Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre. Q 4. Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Q 5. The figure shows a uniform rod of length l whose mass per unit length is ? . What is the gravitational force of the rod on a particle of mass m located a distance d from one end of the rod? Acceleration due to Gravity Q 6. Value of g on the surface of earth is 9.8 m/s 2 . Find its value on the surface of a planet whose mass and radius both are two times that of earth. Q 7. Value of g on the surface of earth is 9.8 m/s 2 . Find its value : (a) at height h = R from the surface, (b) at depth h = R 2 from the surface. (R = radius of earth) Q 8. Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below and above the surface of the earth. Q 9. A body is weighed by a spring balance to be 1000 N at the north pole. How much will it weight at the equator. Account for the earth's rotation only. Q 10. At what rate should the earth rotate so that the apparent g at the equator becomes zero ? What will be the length of the day in this situation ? Q 11. Assuming earth to be spherical, at what height above the north pole, value of g is same as that on the earth's surface at equator ? Gravitational Field Strength and Potential Q 12. Two concentric spherical shells have masses m 1 ,m2 and radii R1, R2 (R1 < R2). Calculate the force exerted by this system on a particle of mass m, if it is placed at a distance 12 (R R ) 2 ? from the centre. Q 13. Two spheres one of mass M has radius R. Another sphere has mass 4M and radius 2R. The centre to centre distance between them is 12R. Find the distance from the centre of smaller sphere where : (a) net gravitational field is zero, (b) net gravitational potential is half the potential on the surface of larger sphere. Q 14. A semicircular wire has a length L and mass M. Find the gravitational field at the centre of the circle. Q 15. A uniform solid sphere of mass M and radius a is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2a. Find the gravitational field at a distance (a) 3 a 2 from the centre, (b) 5 a 2 from the centre. Q 16. The density inside a solid sphere of radius a is given by ? = ?0a/r, where ?0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre. Q 17. A particle of mass m is placed at the centre of a uniform spherical shell of same mass and radius R. Find the gravitational potential at a distance R 2 from the centre. Gravitational Potential Energy Q 18. A rocket is accelerated to speed v 2 gR ? near earth's surface (R =radius of earth). Show that very far from earth its speed will be v 2gR ? . Q 19. Two neutron stars are separated by a distance of 10 10 m. They each have a mass of 10 30 kg and a radius of 10 5 m. They are initially at rest with respect to each other. As measured from the rest frame, how fast are they moving when : (a) their separation has decreased to onehalf its initial value, (b) they are about to collide. Q 20. A projectile is fired vertically from earth's surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of earth will it go ? Q 21. A mass m is taken to a height R from the surface of the earth and then is given a vertical velocity v. Find the minimum value of v, so that mass never returns to the surface of the earth. (Radius of earth is R and mass of the earth M). Q 22. In the figure masses 400 kg and 100 kg are fixed. (a) How much work must be done to move a 1 kg mass from point A to point B ? (b) What is the minimum kinetic energy with which the 1 kg mass must be projected from A to the right to reach the point B ? Planets and Satellites : Kepler's Law Q 23. A sky lab of mass 2 x 10 3 kg is first launched from the surface of earth in a circular orbit of radius 2R and then it is shifted from this circular orbit to another circular orbit of radius 3R. Calculate the energy required: (a) to place the lab in the first orbit, (b) to shift the lab from first orbit to the second orbit. (R = 6400 km, g = 10m/s 2 ) Q 24. Two identical stars of mass M orbit around their centre of mass. Each orbit is circular and has radius R, so that the two stars are always on opposite sides of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) What minimum energy would be required to separate the two stars to infinity ? Q 25. Consider two satellites A and B of equal mass, moving in the same circular orbit of radius r around the earth but in the opposite sense and therefore a collision occurs. (a) Find the total mechanical energy EA+EB of the two satelliteplusearth system before collision. (b) If the collision is completely inelastic, find the total mechanical energy immediately after collision. Describe the subsequent motion of the combined satellite. Q 26. Two satellites A and B revolve around a planet in two coplanar circular orbits in the same sense with radii 10 4 km and 2 x 10 4 km respectively. Time period of A is 28 hours. What is time period of another satellite. Q 27. A satellite of mass 1000 kg is supposed to orbit the earth at a height of 2000 km above the earth's surface. Find (a) its speed in the orbit, (b) its kinetic energy, (c) the potential energy of the earthsatellite system and (d) its time period. Mass of the earth = 6 x 10 24 kg. Q 28. In a certain binary star system, each star has the same mass as our sun. They revolve about their centre of mass. The distance between them is the same as the distance between earth and the sun. What is their period of revolution in years ? Q 29. (a) Does it take more energy to get a satellite upto 1500 km above earth than to put it in circular orbit once it is there. (b) What about 3185 km? (c) What about 4500 km? (Take Re = 6370 km) Solutions 1. = 5.3 × 10 10 m/s 2 = 2.65 × 10 10 m/s 2 2. And the net force is directed along PB. 3. 4. 5. 6. Page 5 EXERCISES For JEE Main Subjective Questions Newton's Law of Gravitation Q 1. Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. Assuming that the only forces acting on the particles are their mutual gravitation, find the initial accelerations of the two particles. Q 2. Three particles A, B and C, each of mass m, are placed in a line with AB = BC = d. Find the gravitational force on a fourth particle P of same mass, placed at a distance d from the particle B on the perpendicular bisector of the line AC. Q 3. Four particles having masses m, 2m, 3m and 4m are placed at the four corners of a square of edge a. Find the gravitational force acting on a particle of mass m placed at the centre. Q 4. Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two. Q 5. The figure shows a uniform rod of length l whose mass per unit length is ? . What is the gravitational force of the rod on a particle of mass m located a distance d from one end of the rod? Acceleration due to Gravity Q 6. Value of g on the surface of earth is 9.8 m/s 2 . Find its value on the surface of a planet whose mass and radius both are two times that of earth. Q 7. Value of g on the surface of earth is 9.8 m/s 2 . Find its value : (a) at height h = R from the surface, (b) at depth h = R 2 from the surface. (R = radius of earth) Q 8. Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below and above the surface of the earth. Q 9. A body is weighed by a spring balance to be 1000 N at the north pole. How much will it weight at the equator. Account for the earth's rotation only. Q 10. At what rate should the earth rotate so that the apparent g at the equator becomes zero ? What will be the length of the day in this situation ? Q 11. Assuming earth to be spherical, at what height above the north pole, value of g is same as that on the earth's surface at equator ? Gravitational Field Strength and Potential Q 12. Two concentric spherical shells have masses m 1 ,m2 and radii R1, R2 (R1 < R2). Calculate the force exerted by this system on a particle of mass m, if it is placed at a distance 12 (R R ) 2 ? from the centre. Q 13. Two spheres one of mass M has radius R. Another sphere has mass 4M and radius 2R. The centre to centre distance between them is 12R. Find the distance from the centre of smaller sphere where : (a) net gravitational field is zero, (b) net gravitational potential is half the potential on the surface of larger sphere. Q 14. A semicircular wire has a length L and mass M. Find the gravitational field at the centre of the circle. Q 15. A uniform solid sphere of mass M and radius a is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2a. Find the gravitational field at a distance (a) 3 a 2 from the centre, (b) 5 a 2 from the centre. Q 16. The density inside a solid sphere of radius a is given by ? = ?0a/r, where ?0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre. Q 17. A particle of mass m is placed at the centre of a uniform spherical shell of same mass and radius R. Find the gravitational potential at a distance R 2 from the centre. Gravitational Potential Energy Q 18. A rocket is accelerated to speed v 2 gR ? near earth's surface (R =radius of earth). Show that very far from earth its speed will be v 2gR ? . Q 19. Two neutron stars are separated by a distance of 10 10 m. They each have a mass of 10 30 kg and a radius of 10 5 m. They are initially at rest with respect to each other. As measured from the rest frame, how fast are they moving when : (a) their separation has decreased to onehalf its initial value, (b) they are about to collide. Q 20. A projectile is fired vertically from earth's surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of earth will it go ? Q 21. A mass m is taken to a height R from the surface of the earth and then is given a vertical velocity v. Find the minimum value of v, so that mass never returns to the surface of the earth. (Radius of earth is R and mass of the earth M). Q 22. In the figure masses 400 kg and 100 kg are fixed. (a) How much work must be done to move a 1 kg mass from point A to point B ? (b) What is the minimum kinetic energy with which the 1 kg mass must be projected from A to the right to reach the point B ? Planets and Satellites : Kepler's Law Q 23. A sky lab of mass 2 x 10 3 kg is first launched from the surface of earth in a circular orbit of radius 2R and then it is shifted from this circular orbit to another circular orbit of radius 3R. Calculate the energy required: (a) to place the lab in the first orbit, (b) to shift the lab from first orbit to the second orbit. (R = 6400 km, g = 10m/s 2 ) Q 24. Two identical stars of mass M orbit around their centre of mass. Each orbit is circular and has radius R, so that the two stars are always on opposite sides of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) What minimum energy would be required to separate the two stars to infinity ? Q 25. Consider two satellites A and B of equal mass, moving in the same circular orbit of radius r around the earth but in the opposite sense and therefore a collision occurs. (a) Find the total mechanical energy EA+EB of the two satelliteplusearth system before collision. (b) If the collision is completely inelastic, find the total mechanical energy immediately after collision. Describe the subsequent motion of the combined satellite. Q 26. Two satellites A and B revolve around a planet in two coplanar circular orbits in the same sense with radii 10 4 km and 2 x 10 4 km respectively. Time period of A is 28 hours. What is time period of another satellite. Q 27. A satellite of mass 1000 kg is supposed to orbit the earth at a height of 2000 km above the earth's surface. Find (a) its speed in the orbit, (b) its kinetic energy, (c) the potential energy of the earthsatellite system and (d) its time period. Mass of the earth = 6 x 10 24 kg. Q 28. In a certain binary star system, each star has the same mass as our sun. They revolve about their centre of mass. The distance between them is the same as the distance between earth and the sun. What is their period of revolution in years ? Q 29. (a) Does it take more energy to get a satellite upto 1500 km above earth than to put it in circular orbit once it is there. (b) What about 3185 km? (c) What about 4500 km? (Take Re = 6370 km) Solutions 1. = 5.3 × 10 10 m/s 2 = 2.65 × 10 10 m/s 2 2. And the net force is directed along PB. 3. 4. 5. 6. 7. (a) (b) 8. ? (Rx)(R + x) 2 = R 3 ? (R 2 x 2 )(R + x)=R 3 or R 3 +R 2 x  x 2 Rx 3 =R 3 or x 3 + x 2 R  xR 2 = 0 or x 2 +Rx  R 2 = 0 9. wp = mg and we = mg  m ? 2 R = 1000 × 0397 = 997 N 10. gapparent = g  ? 2 R = 0 = 16 ? × 10 2 = 5026.5 s = 1.4 hRead More
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